On the convergence of bounded solutions of non homogeneous gradient-like systems

We study the long time behavior of the bounded solutions of non homogeneous gradient-like system which admits a strict Lyapunov function. More precisely, we show that any bounded solution of the gradient-like system converges to an accumulation point as time goes to in nity under some mild hypotheses. As in homogeneous case, the key assumptions for this system are also the angle condition and the Kurdyka-Lojasiewicz inequality. The convergence result will be proved under a L -condition of the perturbation term. Moreover, if the Lyapunov function satis es a Lojasiewicz inequality then the rate of convergence will be even obtained.


Introduction
We are interested in the long time behavior of bounded solutions of the rst order non homogeneous gradient-like system where u ∈ C 1 R + , R N , G ∈ C R N , R N , and f ∈ L 1 R + , R N . The second term f (t) in the right hand side of (1) can be interpreted as a perturbation term for the original equation Roughly speaking, we study in this paper the eect of adding a L 1 forcing term to the equation (2) on the long time behavior of the trajectories u. As in some existing papers on convergence for gradient-like system (2) (see [6] or [15]), we also restrict our study to situations that the system (1) admits a strict Lyapunov function F. That means F (u (t)) is non increasing and the solution u(t) will be constant if F (u (t)) vanishes at some t.
The most simple situation of (2) is the case of gradient system where G = ∇F . This system has been studied by many authors such as Absil & Kurdyka [1], Chill [5], Haraux & Jendoubi [11], [12] or Simon [17]. They have proved that if F satises a Lojasiewicz inequality then the bounded solution converges to an equilibrium as t goes to innity. More general, in a paper of R. Chill et al. [6], the authors gave an abstract result which guarantees that the convergence result also holds for the gradient-like system (2). To obtain the convergence result, they used an additional condition that G, ∇F satisfy an angle condition. In [15] and [16], the authors showed that the hypothesis Lojasiewicz inequality of F can be extended by Kurdyka-Lojasiewicz for convergence result. They even have the rates of convergence if F satises Lo-DOI: http://dx.doi.org/10.25073/jaec.201711.50 VOLUME: 1 | ISSUE: 1 | 2017 | June jasiewicz inequality and G, ∇F satisfy angle and comparability condition.
In the non homogeneous case, recently R. Chill and M. Jendoubi [7] (or Huang and Takac [14]) have shown that any bounded solution of the following gradient system converges to a critical point of F as t tends to innity if f satises for some positive constant µ. This condition shows that the forcing term f (t) quickly decays to zero as t goes to innity. Their results have been generalized to some second order systems in [2], [3], [4], [9] or [10]. Moreover, M. Ghisi et. al. have estimated the decay rates for solutions of semi linear dissipative equations in [8].
Motivated by these works, we establish the convergence results for the rst order nonhomogeneous gradient-like system (1) under a weaker assumption. In the other words, we will show that we can remove the strong assumption of the forcing term (3). In fact, we only need that the forcing term term f belongs to L 1 (R + ) . In particular, our proof seems simpler than the proof in [7] and [14].
In this article, the convergence results will be obtained under the Kurdyka-Lojasiewicz inequality.
The main diculty comes from the generality of non-decreasing function Θ in Kurdyka-Lojasiewicz inequality. To overcome this problem, our idea is to consider Θ satisfying a subadditive property which always holds for the case of Lojasiewicz inequality. Moreover, we also establish a general abstract result for an arbitrary function which are not necessary solutions of (1). Then we apply this abstract result by replacing the energy by a suitale perturbed Lyapunov function. We believe that this general setting enables to quickly check whether convergence properties hold in specic situations.
Our article is organized as follows. In the next section, we present some notations and definitions that we use through the whole of the paper. In the last section, we also establish a general abstract result that we will apply for the main results. Then we prove the convergence of bounded solutions of the nonhomogeneous gradient-like system with the rates of convergence.

Some Denitions
In this paper, to obtain the convergence result, we assume that G and ∇F satisfy the angle condition and F satises the Kurdyka-Lojasiewicz inequality dened below. Denition 1. We say that G and ∇F satisfy the angle condition if there exists a positive number α such that Using the same notation as in [13], we still denote by Q the class of non-decreasing functions Kurdyka-Lojasiewicz inequality at ϕ if there exists σ > 0 and a non-decreasing function Θ ∈ Q such that Throughout this paper, we assume moreover that the function Θ in Kurdyka-Lojasiewicz inequality is subadditive. This means that there exists a constant γ > 0 such that However, the Kurdyka-Lojasiewicz inequality is not sucient to estimate the explicit convergence rate. In this case, we need a Lojasiewicz inequality. Denition 3. We say that the function F satises a Lojasiewicz inequality at ϕ is there exists β, σ > 0 and θ ∈ (0, 1/2] such that, The coecient θ is called a Lojasiewicz exponent.

Remark 1. The Lojasiewicz inequality is a spe-
In particular we note that this function is subadditive by the following lemma.
For every y 0 let us consider the function Computing the rst order derivative of g y at x > 0, we get Note that x → x θ is non-decreasing for every x 0 and θ ∈ (0, 1), so we obtain that g y is nondecreasing. This implies g y (x) g y (0) = 0. The proof is complete.

Convergence Results
Let us study the main result of this paper. We rst establish an abstract convergence result for arbitrary functions which are not necessary solutions of the ordinary dierential equation (1). Then, we prove the convergence result under a L 1 -condition of the forcing term.

3.1.
An abstract convergence result Theorem 1. Let u ∈ C 1 R + , R N be bounded and f ∈ L 1 R + , R N . Assume that the function H ∈ C 1 (R + ) is a non-increasing and H(t) converges to 0 at innity. Assume moreover that there exists a function Θ ∈ Q such that for every t large enough, we have where C is a positive constant. Then u (t) belongs to L 1 (R + ).
In particular, there exists an accumulation point ϕ such that u(t) converges to ϕ as t tends to innity.

Proof.
Let us dene Φ (x) = x 0 1 Θ(s) ds, x 0. Since the function H is non increasing and lim t→∞ H(t) = 0, we deduce that Φ is well-dened and Φ(H(t)) converges to 0 as t goes to innity.
In the other hand, because the function u is bounded, so there exists an accumulation point ϕ of u, it means ϕ ∈ ω [u] := ϕ ∈ R N : ∃t n ↑ such that u (t n ) → ϕ From these above reasons and the hypothesis f ∈ L 1 (R + ), we have that ϕ ∈ ω [u] and Hence, for every ε > 0, we can choose t 0 large enough such that Let us set t 1 = inf {t t 0 : u (t) − ϕ ε}. By (9) and continuity of the function u, we have t 1 t 0 . For every t ∈ [t 0 , t 1 ], using the hypothesis (8), we have the estimation Integrating this estimation on [t 0 , t) for any t ∈ [t 0 , t 1 ], we get It follows from the above estimate that We claim that t 1 = +∞. Indeed, otherwise t 1 < +∞, applying the above estimate for t = t 1 and then using (9), we obtain This contradicts the denition of t 1 . Eventually, the estimate (10) yields that u (t) ∈ L 1 (R + ) and then we deduce u(t) converges to ϕ as t tends to innity by Cauchy criterion.

3.2.
Convergence under a L 1condition of the forcing term We assume that ∇F is bounded from above by a constant K. Let us dene: We prove that the convergence result is obtained if V ∈ L 1 (R + ).

Theorem 2. Let u be a bounded solution
of (1) and f ∈ L 1 (R + ). Assume that G, ∇F satisfy the angle condition (4), F satises the Kurdyka-Lojasiewicz inequality (5) and G (u) C ∇F (u) . If V ∈ L 1 (R + ) then u(t) converges to as t goes to innity.

Proof.
Let us dene H(t) = F (u(t)) + I(t), where We note that since f ∈ L 1 (0, +∞), so ∞ t f (s) ds → 0 as t tends to innity. We deduce I(t) converges to 0 as t goes to innity. Next, we will prove that this function satises the hypotheses of Theorem 1. Indeed, using the angle condition (4), we have 0.
So the function H is non-increasing. Moreover, since u is a bounded solution of (1), which implies that H is bounded from below and there exists an accumulation point ϕ ∈ ω [u]. Therefore, by continuity of F , it follows that H(t) converges to F (ϕ) at innity. Without loss of generality, we may assume that F (ϕ) = 0. In fact, we can dene the energy function H by H(t) = F (u(t)) − F (ϕ) + I(t) in general. Hence, H(t) converges to 0 as t goes to innity. In the other hand, using the angle condition (4), we have Θ(H(t)) .
Next, we will estimate the rate of convergence of bounded solutions of (1). The convergence rate will be depended on the Lojasiewicz exponent θ in (7). For more convenient, we rst state the decay rate for the classical ordinary dierential equation as follows: Lemma 2. Let y be a positive solution of the following ODE: If a > 0 and α 1 the for t large enough, we have Proof.
In the case α = 1, we get that y + ay 0. Writing g(t) := e at y(t), we deduce that g is nonincreasing. Hence, g(t) g(0), for every t 0. We conclude that y(t) g(0)e −at . In the second case α > 1, let us dene g(t) := (y(t)) 1−α . This function satises g (t) (α − 1)a := c which implies g(t) ct for t large enough. It follows y(t) ct −1/(α−1) . Theorem 3. Let u be a bounded solution of (1). Assume that ∇F is bounded from below and G, ∇F satises the angle and comparability condition, this means there exists a constant ν > 0 such that f (s) ds < ∞ for some constant µ > 0 and F satises Lojasiewicz inequality with Lojasiewicz exponent θ ∈ 0, µ 1+µ then u(t) converges to ϕ as t goes to innity. We even have the convergence rate as follows: Proof.

Conclusion
In this article, we establish some convergence results of bounded solutions for non homogeneous rst order gradient-like system under L 1condition of forcing term. We also provide an estimation of convergence rate. The asymptotic behavior of solutions for general second order gradient-like system is still interesting for many people. We hope to study this problem in future works.